Self-attention in vision transformers is often thought to perform perceptual grouping where tokens attend to other tokens with similar embeddings, which could correspond to semantically similar features of an object. However, attending to dissimilar tokens can be beneficial by providing contextual information. We propose to analyze the query-key interaction by the singular value decomposition of the interaction matrix (i.e. ${\textbf{W}_q}^\top\textbf{W}_k$). We find that in many ViTs, especially those with classification training objectives, early layers attend more to similar tokens, while late layers show increased attention to dissimilar tokens, providing evidence corresponding to perceptual grouping and contextualization, respectively. Many of these interactions between features represented by singular vectors are interpretable and semantic, such as attention between relevant objects, between parts of an object, or between the foreground and background. This offers a novel perspective on interpreting the attention mechanism, which contributes to understanding how transformer models utilize context and salient features when processing images.

Computational neuroscience studies have shown that the structure of neural variability to an unchanged stimulus affects the amount of information encoded. Some artificial deep neural networks, such as those with Monte Carlo dropout layers, also have variable responses when the input is fixed. However, the structure of the trial-by-trial neural covariance in neural networks with dropout has not been studied, and its role in decoding accuracy is unknown. We studied the above questions in a convolutional neural network model with dropout in both the training and testing phases. We found that trial-by-trial correlation between neurons (i.e., noise correlation) is positive and low dimensional. Neurons that are close in a feature map have larger noise correlation. These properties are surprisingly similar to the findings in the visual cortex. We further analyzed the alignment of the main axes of the covariance matrix. We found that different images share a common trial-by-trial noise covariance subspace, and they are aligned with the global signal covariance. This evidence that the noise covariance is aligned with signal covariance suggests that noise covariance in dropout neural networks reduces network accuracy, which we further verified directly with a trial-shuffling procedure commonly used in neuroscience. These findings highlight a previously overlooked aspect of dropout layers that can affect network performance. Such dropout networks could also potentially be a computational model of neural variability.

Sensory perception is dramatically influenced by the context. Models of contextual neural surround effects in vision have mostly accounted for Primary Visual Cortex (V1) data, via nonlinear computations such as divisive normalization. However, surround effects are not well understood within a hierarchy, for neurons with more complex stimulus selectivity beyond V1. We utilized feedforward deep convolutional neural networks and developed a gradient-based technique to visualize the most suppressive and excitatory surround. We found that deep neural networks exhibited a key signature of surround effects in V1, highlighting center stimuli that visually stand out from the surround and suppressing responses when the surround stimulus is similar to the center. We found that in some neurons, especially in late layers, when the center stimulus was altered, the most suppressive surround surprisingly can follow the change. Through the visualization approach, we generalized previous understanding of surround effects to more complex stimuli, in ways that have not been revealed in visual cortices. In contrast, the suppression based on center surround similarity was not observed in an untrained network. We identified further successes and mismatches of the feedforward CNNs to the biology. Our results provide a testable hypothesis of surround effects in higher visual cortices, and the visualization approach could be adopted in future biological experimental designs.

Deep convolutional neural networks (CNNs) are becoming increasingly popular models to predict neural responses in visual cortex. However, contextual effects, which are prevalent in neural processing and in perception, are not explicitly handled by current CNNs, including those used for neural prediction. In primary visual cortex, neural responses are modulated by stimuli spatially surrounding the classical receptive field in rich ways. These effects have been modeled with divisive normalization approaches, including flexible models, where spatial normalization is recruited only to the degree that responses from center and surround locations are deemed statistically dependent. We propose a flexible normalization model applied to midlevel representations of deep CNNs as a tractable way to study contextual normalization mechanisms in midlevel cortical areas. This approach captures nontrivial spatial dependencies among midlevel features in CNNs, such as those present in textures and other visual stimuli, that arise from tiling high-order features geometrically. We expect that the proposed approach can make predictions about when spatial normalization might be recruited in midlevel cortical areas. We also expect this approach to be useful as part of the CNN tool kit, therefore going beyond more restrictive fixed forms of normalization.

Spatial context in images induces perceptual phenomena associated with salience and modulates the responses of neurons in primary visual cortex (V1). However, the computational and ecological principles underlying contextual effects are incompletely understood. We introduce a model of natural images that includes grouping and segmentation of neighboring features based on their joint statistics, and we interpret the firing rates of V1 neurons as performing optimal recognition in this model. We show that this leads to a substantial generalization of divisive normalization, a computation that is ubiquitous in many neural areas and systems. A main novelty in our model is that the influence of the context on a target stimulus is determined by their degree of statistical dependence. We optimized the parameters of the model on natural image patches, and then simulated neural and perceptual responses on stimuli used in classical experiments. The model reproduces some rich and complex response patterns observed in V1, such as the contrast dependence, orientation tuning and spatial asymmetry of surround suppression, while also allowing for surround facilitation under conditions of weak stimulation. It also mimics the perceptual salience produced by simple displays, and leads to readily testable predictions. Our results provide a principled account of orientation-based contextual modulation in early vision and its sensitivity to the homogeneity and spatial arrangement of inputs, and lends statistical support to the theory that V1 computes visual salience.

A central hypothesis about early visual processing is that it represents inputs in a coordinate system matched to the statistics of natural scenes. Simple versions of this lead to Gabor-like receptive fields and divisive gain modulation from local surrounds; these have led to influential neural and psychological models of visual processing. However, these accounts are based on an incomplete view of the visual context surrounding each point. Here, we consider an approximate model of linear and non-linear correlations between the responses of spatially distributed Gabor-like receptive fields, which, when trained on an ensemble of natural scenes, unifies a range of spatial context effects. The full model accounts for neural surround data in primary visual cortex (V1), provides a statistical foundation for perceptual phenomena associated with Lis (2002) hypothesis that V1 builds a saliency map, and fits data on the tilt illusion.

Gaussian scale mixture models offer a top-down description of signal generation that captures key bottom-up statistical characteristics of filter responses to images. However, the pattern of dependence among the filters for this class of models is prespecified. We propose a novel extension to the gaussian scale mixture model that learns the pattern of dependence from observed inputs and thereby induces a hierarchical representation of these inputs. Specifically, we propose that inputs are generated by gaussian variables (modeling local filter structure), multiplied by a mixer variable that is assigned probabilistically to each input from a set of possible mixers. We demonstrate inference of both components of the generative model, for synthesized data and for different classes of natural images, such as a generic ensemble and faces. For natural images, the mixer variable assignments show invariances resembling those of complex cells in visual cortex; the statistics of the gaussian components of the model are in accord with the outputs of divisive normalization models. We also show how our model helps interrelate a wide range of models of image statistics and cortical processing.

The misjudgement of tilt in images lies at the heart of entertaining visual illusions and rigorous perceptual psychophysics. A wealth of findings has attracted many mechanistic models, but few clear computational principles. We adopt a Bayesian approach to perceptual tilt estimation, showing how a smoothness prior offers a powerful way of addressing much confusing data. In particular, we faithfully model recent results showing that confidence in estimation can be systematically affected by the same aspects of images that affect bias. Confidence is central to Bayesian modeling approaches, and is applicable in many other perceptual domains. Perceptual anomalies and illusions, such as the misjudgements of motion and tilt evident in so many psychophysical experiments, have intrigued researchers for decades.13 A Bayesian view48 has been particularly influential in models of motion processing, treating such anomalies as the normative product of prior information (often statistically codifying Gestalt laws) with likelihood information from the actual scenes presented. Here, we expand the range of statistically normative accounts to tilt estimation, for which there are classes of results (on estimation confidence) that are so far not available for motion. The tilt illusion arises when the perceived tilt of a center target is misjudged (ie bias) in the presence of flankers. Another phenomenon, called Crowding, refers to a loss in the confidence (ie sensitivity) of perceived target tilt in the presence of flankers. Attempts have been made to formalize these phenomena quantitatively. Crowding has been modeled as compulsory feature pooling (ie averaging of orientations), ignoring spatial positions.9, 10 The tilt illusion has been explained by lateral interactions11, 12 in populations of orientationtuned units; and by calibration.13 However, most models of this form cannot explain a number of crucial aspects of the data. First, the geometry of the positional arrangement of the stimuli affects attraction versus repulsion in bias, as emphasized by Kapadia et al14 (figure 1A), and others.15, 16 Second, Solomon et al. recently measured bias and sensitivity simultaneously.11 The rich and surprising range of sensitivities, far from flat as a function of flanker angles (figure 1B), are outside the reach of standard models. Moreover, current explanations do not offer a computational account of tilt perception as the outcome of a normative inference process. Here, we demonstrate that a Bayesian framework for orientation estimation, with a prior favoring smoothness, can naturally explain a range of seemingly puzzling tilt data. We explicitly consider both the geometry of the stimuli, and the issue of confidence in the esti- (A) 6 5 4 3 2 1 0 -1 -2

In the analysis of natural images, Gaussian scale mixtures (GSM) have been used to account for the statistics of (cid:2)lter responses, and to inspire hi- erarchical cortical representational learning schemes. GSMs pose a crit- ical assignment problem, working out which (cid:2)lter responses were gen- erated by a common multiplicative factor. We present a new approach to solving this assignment problem through a probabilistic extension to the basic GSM, and show how to perform inference in the model using Gibbs sampling. We demonstrate the ef(cid:2)cacy of the approach on both synthetic and image data. Understanding the statistical structure of natural images is an important goal for visual neuroscience. Neural representations in early cortical areas decompose images (and likely other sensory inputs) in a way that is sensitive to sophisticated aspects of their probabilistic structure. This structure also plays a key role in methods for image processing and coding. A striking aspect of natural images that has re(cid:3)ections in both top-down and bottom-up modeling is coordination across nearby locations, scales, and orientations. From a top- down perspective, this structure has been modeled using what is known as a Gaussian Scale Mixture model (GSM).1(cid:150)3 GSMs involve a multi-dimensional Gaussian (each di- mension of which captures local structure as in a linear (cid:2)lter), multiplied by a spatialized collection of common hidden scale variables or mixer variables(cid:3) (which capture the coordi- nation). GSMs have wide implications in theories of cortical receptive (cid:2)eld development, eg the comprehensive bubbles framework of Hyv¤arinen.4 The mixer variables provide the top-down account of two bottom-up characteristics of natural image statistics, namely the ‘bowtie’ statistical dependency,5,6 and the fact that the marginal distributions of receptive (cid:2)eld-like (cid:2)lters have high kurtosis.7,8 In hindsight, these ideas also bear a close relation- ship with Ruderman and Bialek’s multiplicative bottom-up image analysis framework9 and statistical models for divisive gain control.6 Coordinated structure has also been addressed in other image work,10(cid:150)14 and in other domains such as speech15 and (cid:2)nance.16 Many approaches to the unsupervised speci(cid:2)cation of representations in early cortical areas rely on the coordinated structure.17(cid:150)21 The idea is to learn linear (cid:2)lters (eg modeling simple cells as in22,23), and then, based on the coordination, to (cid:2)nd combinations of these (perhaps non-linearly transformed) as a way of (cid:2)nding higher order (cid:2)lters (eg complex cells). One critical facet whose speci(cid:2)cation from data is not obvious is the neighborhood arrangement, ie which linear (cid:2)lters share which mixer variables. (cid:3)Mixer variables are also called mutlipliers, but are unrelated to the scales of a wavelet. Here, we suggest a method for (cid:2)nding the neighborhood based on Bayesian inference of the GSM random variables. In section 1, we consider estimating these components based on information from different-sized neighborhoods and show the modes of failure when inference is too local or too global. Based on these observations, in section 2 we propose an extension to the GSM generative model, in which the mixer variables can overlap prob- abilistically. We solve the neighborhood assignment problem using Gibbs sampling, and demonstrate the technique on synthetic data. In section 3, we apply the technique to image data. 1 GSM inference of Gaussian and mixer variables In a simple, n-dimensional, version of a GSM, (cid:2)lter responses l are synthesized y by mul- tiplying an n-dimensional Gaussian with values g = fg1 : : : gng, by a common mixer variable v. (1) We assume g are uncorrelated ((cid:27)2 along diagonal of the covariance matrix). For the ana- lytical calculations, we assume that v has a Rayleigh distribution:

Spike-triggered averaging techniques are effective for linear characterization of neural responses. But neurons exhibit important nonlinear behaviors, such as gain control, that are not captured by such analyses. We describe a spike-triggered covariance method for retrieving suppressive components of the gain control signal in a neuron. We demonstrate the method in simulation and on retinal ganglion cell data. Analysis of physiological data reveals significant suppressive axes and explains neural nonlinearities. This method should be applicable to other sensory areas and modalities.

We explore the statistical properties of natural sound stimuli pre(cid:173) processed with a bank of linear filters. The responses of such filters exhibit a striking form of statistical dependency, in which the response variance of each filter grows with the response amplitude of filters tuned for nearby frequencies. These dependencies may be substantially re(cid:173) duced using an operation known as divisive normalization, in which the response of each filter is divided by a weighted sum of the recti(cid:173) fied responses of other filters. The weights may be chosen to maximize the independence of the normalized responses for an ensemble of natu(cid:173) ral sounds. We demonstrate that the resulting model accounts for non(cid:173) linearities in the response characteristics of the auditory nerve, by com(cid:173) paring model simulations to electrophysiological recordings. In previous work (NIPS, 1998) we demonstrated that an analogous model derived from the statistics of natural images accounts for non-linear properties of neurons in primary visual cortex. Thus, divisive normalization appears to be a generic mechanism for eliminating a type of statistical dependency that is prevalent in natural signals of different modalities. Signals in the real world are highly structured. For example, natural sounds typically con(cid:173) tain both harmonic and rythmic structure. It is reasonable to assume that biological auditory systems are designed to represent these structures in an efficient manner [e.g., 1,2]. Specif(cid:173) ically, Barlow hypothesized that a role of early sensory processing is to remove redundancy in the sensory input, resulting in a set of neural responses that are statistically independent. Experimentally, one can test this hypothesis by examining the statistical properties of neural responses under natural stimulation conditions [e.g., 3,4], or the statistical dependency of pairs (or groups) of neural responses. Due to their technical difficulty, such multi-cellular experiments are only recently becoming possible, and the earliest reports in vision appear consistent with the hypothesis [e.g., 5]. An alternative approach, which we follow here, is to develop a neural model from the statistics of natural signals and show that response properties of this model are similar to those of biological sensory neurons. A number of researchers have derived linear filter models using statistical criterion. For vi(cid:173) sual images, this results in linear filters localized in frequency, orientation and phase [6, 7]. Similar work in audition has yielded filters localized in frequency and phase [8]. Although these linear models provide an important starting point for neural modeling, sensory neu(cid:173) rons are highly nonlinear. In addition, the statistical properties of natural signals are too complex to expect a linear transformation to result in an independent set of components. Recent results indicate that nonlinear gain control plays an important role in neural pro(cid:173) cessing. Ruderman and Bialek [9] have shown that division by a local estimate of standard deviation can increase the entropy of responses of center-surround filters to natural images. Such a model is consistent with the properties of neurons in the retina and lateral genicu(cid:173) late nucleus. Heeger and colleagues have shown that the nonlinear behaviors of neurons in primary visual cortex may be described using a form of gain control known as divisive normalization [10], in which the response of a linear kernel is rectified and divided by the sum of other rectified kernel responses and a constant. We have recently shown that the responses of oriented linear filters exhibit nonlinear statistical dependencies that may be substantially reduced using a variant of this model, in which the normalization signal is computed from a weighted sum of other rectified kernel responses [11, 12]. The resulting model, with weighting parameters determined from image statistics, accounts qualitatively for physiological nonlinearities observed in primary visual cortex. In this paper, we demonstrate that the responses of bandpass linear filters to natural sounds exhibit striking statistical dependencies, analogous to those found in visual images. A divisive normalization procedure can substantially remove these dependencies. We show that this model, with parameters optimized for a collection of natural sounds, can account for nonlinear behaviors of neurons at the level of the auditory nerve. Specifically, we show that: 1) the shape offrequency tuning curves varies with sound pressure level, even though the underlying linear filters are fixed; and 2) superposition of a non-optimal tone suppresses the response of a linear filter in a divisive fashion, and the amount of suppression depends on the distance between the frequency of the tone and the preferred frequency of the filter. 1 Empirical observations of natural sound statistics The basic statistical properties of natural sounds, as observed through a linear filter, have been previously documented by Attias [13]. In particular, he showed that, as with visual images, the spectral energy falls roughly according to a power law, and that the histograms of filter responses are more kurtotic than a Gaussian (i.e., they have a sharp peak at zero, and very long tails). Here we examine the joint statistical properties of a pair of linear filters tuned for nearby temporal frequencies. We choose a fixed set of filters that have been widely used in mod(cid:173) eling the peripheral auditory system [14]. Figure 1 shows joint histograms of the instanta(cid:173) neous responses of a particular pair of linear filters to five different types of natural sound, and white noise. First note that the responses are approximately decorrelated: the expected value of the y-axis value is roughly zero for all values of the x-axis variable. The responses are not, however, statistically independent: the width of the distribution of responses of one filter increases with the response amplitude of the other filter. If the two responses were statistically independent, then the response of the first filter should not provide any information about the distribution of responses of the other filter. We have found that this type of variance dependency (sometimes accompanied by linear correlation) occurs in a wide range of natural sounds, ranging from animal sounds to music. We emphasize that this dependency is a property of natural sounds, and is not due purely to our choice of lin(cid:173) ear filters. For example, no such dependency is observed when the input consists of white noise (see Fig. 1). The strength of this dependency varies for different pairs of linear filters . In addition, we see this type of dependency between instantaneous responses of a single filter at two